Abstract: Let be r-dimensional real projective space with r odd, and let be the group of orientation preserving diffeomorphisms factored by the normal subgroup of those concordant (= pseudoisotopic) to the identity. The main theorem of this paper is that for the group is isomorphic to the homotopy group , where with and . The function is denned by . The theorem is proved by introducing a cobordism version of the mapping torus construction; this mapping torus construction is a homomorphism for and a suitable Lashof cobordism group. It is shown that t is an isomorphism onto the torsion subgroup , and that this subgroup is isomorphic to as above. Then one reads off from Mahowald's tables of that and .